Higher Reciprocity Laws and Ternary Linear Recurrence Sequences

TL;DR

This paper investigates prime splitting in non-abelian fields related to cubic polynomials, establishing divisibility properties of ternary recurrence sequences and analyzing solutions modulo primes, extending previous research.

Contribution

It introduces new divisibility results for ternary recurrence sequences and simplifies the analysis of solutions modulo primes in the context of non-abelian splitting fields.

Findings

Prime numbers splitting completely in certain non-abelian fields are characterized.

Divisibility properties of associated ternary recurrence sequences by primes are established.

New results on solutions of recurrence characteristic equations modulo primes are proved.

Abstract

We describe the set of prime numbers splitting completely in the non-abelian splitting field of certain monic irreducible polynomials of degree three. As an application we establish some divisibility properties of the associated ternary recurrence sequence by primes $p$, thus greatly extending recent work of Evink and Helminck and of Faisant. We also prove some new results on the number of solutions of the characteristic equation of the recurrence sequence modulo $p,$ extending and simplifying earlier work of Zhi-Hong Sun (2003).